Abstract: This work describes the mathematics needed to implement Elliptic Curve Cryptography (ECC) with special attention to its implementation in Galois Field. Here the functionality of ECC, its advantages and challenges over other cryptosystems are also explained. Comparison with other cryptographic systems will also be undertaken based on aspects such as efficiency, size of the key needed to attain a certain level of security (this has implications on computational costs and time), known and probable attacks, current and predicted future attacks (based on the current growth of technology) and their prevention techniques. ECC reliability will also be looked into.